Calculus:  Part 2 -- Integrals and Applications

Comp110 Course Module 3b

Important Note: Be sure and read the last part of this assignment before you start working.  At the end of this assignment  you will be asked to write and turn in a report on this unit.

Integrals

The derivative with respect to a variable t maps a function to a function.  The inverse map is often referred to as the anti-derivative. Since the derivative of a constant is zero, the inverse map is not single valued.  For example, the anti-derivative of 2t is (t^2)+c where c is a constant of integration.  The anti-derivative (or indefinite integral) of a function v[t]is denoted by

∫v[t]dt.

The value of the indefinite integral at a particular parameter value t is nebulous due to the constant c in the definition of the anti-derivative.  However, the difference in value of the indefinite integral at two parameter values a and b is independent of the constant c.  This expression is referred to as the definite integral of v[t] evaluated on the interval [a,b] and is denoted by

(∫_a^b)v[t]dt.

The definite integral has a nice geometric interpretation.  It represents the area under the graph of v[t] on the interval [a,b]. This area is given by a limit -- the limit of the area of the discrete rectangles that approximate this continuous area.  This limit is written as

(∫_a^b)v[t]dt = (Underscript[Lim, n->∞])(∑_(i=0)^n)v[(Subscript[t, i])]δt.

where δt is the width of the ith rectangle and v[(Subscript[t, i])] is the height of the ith rectangle.  Below is an example of using this idea to compute the area under the graph f[t]=t-(t^3) for the interval [-1,1] using twenty rectangles of width .1.

Newton showed that this limit is given by the anti-derivative.  Intuitively Newton's argument is quite simple.  If you look at how the area under a curve v[t] changes as you sweep a vertical line to the right, then you seen that the instantaneous change in the area under the curve is just the value of the function v[t].  Thus the derivative of the area under the curve is the function, so the anti-derivative of the function is equal to the area under the curve.  This result is known as the Fundamental Theorem of the Calculus and relates integration (area) to differentation (instantaneous rates of change).

Problems

Applications

Maximums and Minimums

One nice application of derivatives is in computing the maxima and minima of a function.  In this section, we lead you through this process.

Problems

Differential Equations

A differential equation of order n relates a function f[t] to various combinations of its derivatives of up to order n,

(∑_(i=0)^n)(Subscript[c, i])Derivative[i][f][t]==0.

(Note that Derivative[i][f] denote the ith Derivative of f here. )  Differential equations model a wide range of physical and natural phenomena.  In the rest of this module, we consider a simple differential equation that models population growth.

Given an adequate set of resources, a reproducing population tends to grow as a function of time t.  In particular, the rate of growth of a population f[t] is typically proportional to f[t]. Therefore, the function f[t] satisfies the differential equation

Derivative[1][f][t]==k*f[t],

where k is constant that lies in the range 0<k<1 and that depends on the reproduction characteristics of the population.  In the next few problems, we will develop a discrete method for computing an approximate solution to this differential equation for the case where k==1.

Problems